Back in 1947 all UN security council deliberations on Palestine (as it was then known) were attended by representatives of the Palestinian-Arabs and the Palestinian-Jews. They also had little signs and the Arab's sign said "Arab High Committee" and the Jew's sign said "Jewish Agency for Palestine".
At some point in 1948 this changed, the Palestinians ceased to attend the meetings and were represented instead by the Egyptian UN representative.
In July 1948, the Ukrainian representative, Dmitry Manuilsky, was the chair of the Security Council. The security council chair is held for a month and some shenanigan by Stalin had enabled the USSR to wangle three seats in the General Assembly, one for the USSR, one for the Ukraine and one for Bayelorussia. in the late forties' the Ukraine also sat in the Security Council. The chair rotates among Security Council members and each country holds the chair one month in alphabetic order, in 1947-8 the Ukraine held the chair every July.
As chair of the Security Council the Ukrainian representative, who I recall seemed to have arrived early, changed the sign saying Jewish Agency, to read simply "Israel".
As it happens the Palestinian observer at the UN is actually the representative of the PLO: In 1974, the General Assembly granted observer status to the PLO, and it was only in 1988, following a declaration of Palestinian Independence (made in Algeria...) that the General Assembly decided that the sign should say "Palestine", although the representative is still appointed by the PLO. Hamas, even though it won the only full free Palestinian elections ever held, cannot appoint the representative.
The chair of the UN security council in September (when the Palestinians submitted their request) was Lebanon. Israel has never been a member of the UN Security Council and is the only long-term UN member which has never sat on it.
Incidentally, while it is true that Palestine and Israel have taken up a lot of Security Council time, if you check the records for the last year you can see that the Middle East in general occupies an awful lot of security council time.